Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators

Abstract

The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

“…It remains another condition that is the discriminant must be positive to gain real roots of (64). It is easy to show that the discriminant Δ ¼…”

“…Ex9: The system of two coupled Van der Pol oscillators is one of the canonical models exhibiting the mutual synchronization behavior. 64 Consider the following coupled Duffing-Van der Pol oscillator…”

An upper limit for slow-earthquake zones: self-oscillatory behavior through the Hopf bifurcation mechanism from a spring-block model under lubricated surfaces

“…The local and asymptotic stability of x * is analyzed with the indirect method of Lyapunov that consists of the analysis of the eigenvalues of the Jacobian matrix from the linearized system of Eq. (5) around the stationary solution (Khalil, 1996). Let x * be locally asymptotically stable, i. e., every solution of the system ϕ t (x 0 ) = (θ (t), u(t), v(t)); starting near to the stationary solution, it remains at the surrounding of x * all the time, and eventually the solution converges to x * (convergence to frictional stability).…”

Master-slave synchronization in the Rayleigh and Duffing oscillators via elastic and dissipative couplings

“…As far as the coupling between the Rayleigh and Duffing oscillators is referred, we can mention three different couplings, namely: gyroscopic, dissipative and elastic [29][30][31][32][33][34]. Among the diverse way of coupling, the most used are the elastic and dissipative ones [34,35,36]. In a previous work [34], it is analyzed a different approach of synchronizing two distinct oscillators of low-dimensional, using the aforementioned couplings.…”